Kent
> You should google the "Delphi Method" or analyzing data. It seems you're missing a critical component for this method.
Actually, this is a fairly clean statistical exercise that gives a result that is quite realistic, even if many surveyors don't expect it or want to believe otherwise. I'll give everyone a chance to work the problem. There's no hurry.
When ability to remember the right words to look up in the index of a statistics book fail, there is always virtual experimentation.
Using Excel, define two random variables, u and v, where u is the distance in the x direction from the lost point to the reset point, which is negative if the reset point is west of the lost point. The definition of v is similar, except in the north direction. The standard deviation of u and v is about 0.282 as Kent derived. So I just generated 200 sets of u and v, found the distance from (0, 0) to (u, v) for each trial, and plotted a histogram of the results. The histogram was indeed non-normal. Empirically, the value for which 100 values were greater and 100 values were less was 0.033 ft.
I believe a more realistic model of a process such as RTK might be a distance r, which has a standard deviation of, for example, 0.02 ft, and a uniformly distributed random direction. I believe this would give slightly different results.
Kent
So you're not going to look up alternative methods to quantify data? Seems odd for you, but okay brother! 🙂
>The standard deviation of u and v is about 0.282 as Kent derived. So I just generated 200 sets of u and v, found the distance from (0, 0) to (u, v) for each trial, and plotted a histogram of the results. The histogram was indeed non-normal. Empirically, the value for which 100 values were greater and 100 values were less was 0.033 ft.
If you generated the trials correctly, the answer to the problem is the bin of the histogram that contains the most members (assuming that all of the bins are the same interval). That is the maximum probability error, the expected error.
> I believe a more realistic model of a process such as RTK might be a distance r, which has a standard deviation of, for example, 0.02 ft, and a uniformly distributed random direction.
The usual way of modeling this would be as standard deviations of the x and y components with their covariances. I assumed for the purposes of this that the N and E values that the positioning system returned were not highly correlated (a generous assumption).
Kent
> So you're not going to look up alternative methods to quantify data?
The so-called "Delphi Method" when applied to subjects like this would be reverting to a faith-based approach that no professional land surveyor would want to be standing near when it blows up.
The problem I posed is a fairly straight-forward statistics problem, so any "expert" who comes up with a different answer probably isn't an expert. :>
Kent
Actually, no. Did you look it up? Evidently not.
This method is widely used by folks with more brains that you and me and much more money on making huge decisions when evaluating data.
Kent
> This method is widely used by folks with more brains that you and me and much more money on making huge decisions when evaluating data.
Kris, of course I looked it up. It obviously has zero application to this problem.
Kent
I totally disagree. The premise behind the Delphi Method is, an application of quantifiable data in conjunction with qualifiable data, for estimating and decision making.
To bring it around to this subject, no one disagrees with your mathematical approach to the issue. However, it's quite difficult to argue with results. So, then either one of two things has happened. Either (A), you did the math wrong (which I highly doubt) or (B) there are more things at play here than your mathematical model can illustrate.
I've said it many times, our RTK gear gets 0.03' reliability most of the time. It's true we "find" (as you said) bad answers, but checking is necessary, as I'm sure you agree.
As always with us surveyors, the proof is in the dirt. So, I'm quite capable of sitting here and telling you, your math is right, and your conclusions are wrong. Why, I've witnessed it and tested it SO many times, it's really just not worth mentioning. I have ZERO problems setting pairs, traversing, and closing into pairs, without having them least square adjusted into some model. They are good enough (most of the time) to stand on their own.
Am I advocating 1 shot RTK ideas, of course not. Because I know your math is right, I'm advocating not throwing the gear away because it's not static and used in Star Net, but that it should be used with care and checking. I've run between enough pairs and checked enough to say, it's so close, you won't be able to tell if it's the gun, the instrument operator, or the GPS points, when done correctly.
That's the qualitative side of it. It's not faith based at all, in fact, it's a quantifiable experience from using the gear.
I believe you said it best, and I may be paraphrasing here "You don't need a degree in metallurgy to use a hammer" (I have to admit, I'm quite fond of that saying and have used it many many times, so I hope you don't mind adding it to my vernacular). And as you've stated OVER AND OVER, you don't use RTK gear. I think you'll find, WHEN you bite off into it, that you've been missing a huge productivity multiplier and can really work it into your everyday work (most of the time).
It's been fun. I enjoy finally getting you to say something. 🙂 Trust me, it wouldn't be so popular, if only your math was at play and correct. No one would use it.
Kent
> However, it's quite difficult to argue with results. So, then either one of two things has happened. Either (A), you did the math wrong (which I highly doubt) or (B) there are more things at play here than your mathematical model can illustrate.
Kris, the problem is in fact a purely statistical one. Are you saying that you don't think that random RTK errors are normally distributed? What is the basis for that assertion, if so?
Kent
Now where did I say that? Nowhere. No, what I said was, the Delphi method incorporates both quantifiable (numbers) and qualifiable (experience) into the mix. Refusing to look at one, well that's a little myopic.
You gotta give the 5700 a spin with the RTK turned on brother! You're gonna love it!
BTW I just noticed a typo. Instead of :
D = SQRT{[ n(1)- n(2)]^2 + [ e(1)+ e(2) ]^2 },
make that:
D = SQRT{[ n(1)- n(2)]^2 + [ e(1)- e(2) ]^2 }
I'm sure that anyone working on this problem wasn't thrown off track by that detail, but for the record there it is.
Kent
> Now where did I say that? Nowhere. No, what I said was, the Delphi method incorporates both quantifiable (numbers) and qualifiable (experience) into the mix. Refusing to look at one, well that's a little myopic.
Kris, just think about what you're saying for a few seconds. The thrust of what you're wanting to argue is that you somehow "disagree" with numerical analysis because of some experience you had. Well, the only sort of experience you could have had that would cast the problem as I've posed it in any doubt is if you had an experience that somehow convinced you that random RTK GPS errors aren't normally distributed. Now you say that you don't dispute that. So, you're left with basically ... nothing, right?
The way that the RTK enthusiasts are running for the hills on this little problem is eyebrow raising, to say the least. Do you somehow suspect that the answer is going to be one that you don't want to know about?
You're likely to repeat within .04' but both measurements can be out 8' or more.
Interesting, and reasonable discussion overall, but I wouldn't agree with this statement.
If you initialize twice and occupy, I'd bet a month's paycheck you'll never initialize incorrectly the exact same way and have the two be within .04' of each other. Just doesn't happen.
> You're likely to repeat within .04' but both measurements can be out 8' or more.
It has nothing to do with incorrect initialization. It has to do with local multipath. And yes, I'll bet ya a month's pay that I can prove you can have to measurements within .04 of each other and still be many feet away from the true value.
Is it a bet? Shall we say, $1000?
> Maybe running from the same point being made over and over with nothing new, and with a side dose of obtuseness we don't need or deserve.
Well, why do you suppose that no one has given the answer, then, if this isn't actually something novel that they haven't thought carefully about? The poster "ashton" got close with a viable method of solution, but didn't recognize the answer in the data produced by the Monte Carlo method he employed.
This particular problem as posed assumed that the coordinates of the marker as found and of the new replacement as set had random, normally distributed errors with a standard error of +/-0.02 ft. The nature of the problem is such that if you substitute a standard error of +/-0.015 ft. or +/-0.03 ft., the most probable error of the replaced mark, the expected error, is just decreased or increased in the same ratio as the new s.e. to 0.02 ft. The fact that the most probable error isn't zero and is somewhat larger than most RTK users would expect doesn't go away.
Mmmm, modeled multipath. I succumb, and I've done it in a lab type of environment. However, I don't think that's what we're talking about here because if you're trying to use GPS in an environment where multi-path is that clean and consistent, you shouldn't be using GPS in the first place.
> It has nothing to do with incorrect initialization. It has to do with local multipath. And yes, I'll bet ya a month's pay that I can prove you can have to measurements within .04 of each other and still be many feet away from the true value.
>
> Is it a bet? Shall we say, $1000?
There is something that bothers me about the original statement of the problem:
"What is the most probable error that you will make in remarking the corner? That is, what is the most likely distance that the new marker you set with, say, network RTK that can measure coordinates with standard errors of, say, +/-0.02 ft. will be from the original position of the marker (of which no trace remains other than the previously determined coordinates)?"
In pure mathematics, all the distances will be real numbers with an infinite number of digits after the decimal, so the probability of any distance exactly equaling any value that might be proposed as the answer is 0. Or if the proposed answer is chosen to match one of the distances, there will only be one value that exactly matches.
Of course, real GPS units will have limited bits to store and report results, so the probability of an answer matching more than one distance becomes non-zero, but I still don't think that is what was intended. Perhaps the mean distance is really what Kent wants?
> It has nothing to do with incorrect initialization. It has to do with local multipath. And yes, I'll bet ya a month's pay that I can prove you can have to measurements within .04 of each other and still be many feet away from the true value.
>
> Is it a bet? Shall we say, $1000?
Maybe I'm wrong, and am willing to learn something here.
Are you saying you have taken a "fixed" shot, flipped the head over (lose initialization), regain "fix", and then got the same bad reading twice?
This has never happened to me.
> In pure mathematics, all the distances will be real numbers with an infinite number of digits after the decimal, so the probability of any distance exactly equaling any value that might be proposed as the answer is 0. Or if the proposed answer is chosen to match one of the distances, there will only be one value that exactly matches.
In this case, the most probable or expected value is the value at the peak in the distribution of the errors, just as the most probable or expected value of a normally distributed set of errors with zero mean is the value at the peak of that distribution. The outcomes cluster around that probable value.